Monte Carlo approximation through Gibbs output in generalized linear mixed models

Geyer (J. Roy. Statist. Soc. 56 (1994) 291) proposed Monte Carlo method to approximate the whole likelihood function. His method is limited to choosing a proper reference point. We attempt to improve the method by assigning some prior information to the parameters and using the Gibbs output to evaluate the marginal likelihood and its derivatives through a Monte Carlo approximation. Vague priors are assigned to the parameters as well as the random effects within the Bayesian framework to represent a non-informative setting. Then the maximum likelihood estimates are obtained through the Newton Raphson method. Thus, out method serves as a bridge between Bayesian and classical approaches. The method is illustrated by analyzing the famous salamander mating data by generalized linear mixed models.